Complete-Case Restrictions With Baseline Covariates

Why condition the restriction on baseline covariates?

A complete-case restriction borrows information about a missing outcome from participants who have complete outcome records. But complete cases and participants with missing outcomes can have different baseline covariate distributions. If outcome distributions vary with those covariates, simply borrowing the pooled complete-case distribution can give the dropout group the wrong distribution. Conditioning on baseline covariates makes explicit which comparisons the restriction assumes are appropriate.

The restriction is a modeling assumption about an unobserved outcome distribution. It is not verified merely because baseline covariates were measured, or because a study was randomized. Its role is to fill the identification gap for outcomes that were not observed. The earlier material on the complete-case restriction establishes the basic two-pattern setup; here the central question is how that restriction changes when baseline information is included in the conditioning set.

Definition: Consider a two-wave setting in which the second-wave outcome $Y_2$ is missing for some participants. Let $C$ denote the complete-case pattern, and let $D$ denote a pattern in which $Y_2$ is missing but the earlier outcome history is observed. Let $X$ be a vector of baseline covariates and $H$ the observed outcome history available before $Y_2$ is missing. A covariate-conditional complete-case restriction is $$ F_{Y_2\mid X,H,D}(y\mid x,h) = F_{Y_2\mid X,H,C}(y\mid x,h) $$ for every relevant $y$ and every covariate-history stratum $(x,h)$ represented in pattern $D$. Here $F$ denotes a conditional cumulative distribution function.

The equality concerns the entire conditional distribution, not just its mean or a particular outcome probability. In a setting with several dropout patterns, the same idea can be stated for each pattern with a missing outcome: condition on baseline covariates and the observed history for that pattern, then specify which complete-case conditional distribution is borrowed. If treatment assignment or another design variable is relevant, the restriction should be stated within treatment group or include that variable in the conditioning set.

The conditioning set must be defined before applying the restriction. For example, if the observed first-wave outcome is relevant to the second-wave outcome, then $H$ should include that outcome; conditioning only on baseline $X$ would make a different and generally stronger assumption. Conversely, adding a covariate changes the restriction: the complete cases and dropouts are now compared within finer strata rather than as pooled groups.

How the borrowed distribution is assembled

The restriction supplies a conditional distribution within each stratum. To obtain a distribution for all members of the dropout pattern, those stratum-specific distributions must be combined using the covariate and history distribution of the dropout pattern. For discrete $X$ and $H$, the law of total probability gives

F_{Y_2\mid D}(y) = \sum_{x,h} F_{Y_2\mid X,H,D}(y\mid x,h)\,P(X=x,H=h\mid D).

Applying the restriction yields the borrowed-distribution formula

F_{Y_2\mid D}(y) = \sum_{x,h} F_{Y_2\mid X,H,C}(y\mid x,h)\,P(X=x,H=h\mid D).

For continuous covariates, the sum is replaced by an integral over the dropout pattern’s covariate-history distribution. This calculation has two distinct ingredients: conditional outcome distributions estimated among complete cases, and stratum weights describing the dropout pattern. Using the complete cases’ weights instead would reconstruct the complete-case distribution, not the distribution for the dropout group.

Worked Example: Stating the Restriction Within Covariate Strata

Suppose $X$ indicates whether a participant has a baseline risk factor (0 = no, 1 = yes). The first-wave outcome $H$ and $X$ are observed in both patterns. For this illustration, consider participants with $H=0$ in one treatment group, so the restriction is being applied within that observed-history and treatment stratum. The second-wave outcome $Y_2$ is binary, with 1 denoting an event.

Among complete cases with $H=0$, suppose the hypothetical data are:

Baseline risk factor $X$	Complete cases	Events $Y_2=1$	Observed event proportion
0	20	4	$4/20=0.20$
1	80	48	$48/80=0.60$

The restriction states that among dropouts with $X=0,H=0$, the missing-outcome event probability is borrowed as $0.20$; among dropouts with $X=1,H=0$, it is borrowed as $0.60$. In notation, for these strata, $P(Y_2=1\mid X=x,H=0,D)=P(Y_2=1\mid X=x,H=0,C)$. This statement requires both strata to be represented among dropouts and complete cases. It does not assert that the two risk-factor groups have the same outcome distribution.

Worked Example: Weighting by the Dropout Covariate Distribution

Continue with the hypothetical conditional event proportions above. Suppose the 100 participants in pattern $D$ with $H=0$ comprise 80 with $X=0$ and 20 with $X=1$. Under the conditional complete-case restriction, the event probability for the dropout pattern is

\begin{aligned} P(Y_2=1\mid D,H=0) &=0.20\,P(X=0\mid D,H=0)+0.60\,P(X=1\mid D,H=0)\\ &=0.20\left(\frac{80}{100}\right)+0.60\left(\frac{20}{100}\right)\\ &=0.16+0.12=0.28. \end{aligned}

Thus the borrowed event probability is $0.28$, or 28% of this dropout stratum. The arithmetic uses the dropout group’s covariate composition: 80% have $X=0$ and 20% have $X=1$. By contrast, pooling the complete cases gives $(4+48)/(20+80)=52/100=0.52$. That pooled proportion weights the conditional probabilities by the complete cases’ composition, which is 20% with $X=0$ and 80% with $X=1$. It does not give the distribution implied for this dropout group.

The difference between 0.28 and 0.52 is not an arithmetic correction to the complete-case estimate. It arises because the conditional outcome probabilities differ by $X$, and the two patterns have different $X$ distributions. The conditional restriction borrows the stratum-specific probabilities; the target pattern’s covariate distribution determines how they are combined.

Derivation and assumptions

For discrete strata, the mixture formula follows directly by partitioning the event $Y_2\leq y$ according to the values of $X$ and $H$. Because the strata are mutually exclusive and exhaustive within $D$,

\begin{aligned} P(Y_2\leq y\mid D) &=\sum_{x,h}P(Y_2\leq y,X=x,H=h\mid D)\\ &=\sum_{x,h}P(Y_2\leq y\mid X=x,H=h,D)P(X=x,H=h\mid D). \end{aligned}

The restriction replaces each conditional probability in the last line by $P(Y_2\leq y\mid X=x,H=h,C)$. This yields the borrowed-distribution formula. The argument uses the law of total probability plus the stated restriction; it does not show that the restriction is true.

Applying this result requires several conditions. First, the pattern and conditioning variables must be defined consistently: $X$ is baseline information, and $H$ contains the observed history being conditioned on. Second, for every stratum with positive probability in $D$, there must be complete cases in that stratum, or a justified model for extending their conditional distribution there. This is the overlap, or positivity, requirement. Third, the complete-case conditional distribution must be estimable with adequate precision. Finally, the equality of conditional distributions across patterns must be defensible as an assumption for the scientific setting. The observed data cannot directly test equality for the missing $Y_2$ values in pattern $D$.

When $X$ or $H$ is continuous, exact matching on every value is usually infeasible. A model or smoothing procedure may then estimate $F_{Y_2\mid X,H,C}$. That introduces additional modeling choices: the functional form, interactions, and tail behavior must be adequate for the covariate region occupied by dropouts. A fitted conditional distribution should not be treated as reliable extrapolation simply because a model returns a prediction.

Worked Example: Detecting a Stratum With No Complete-Case Support

Suppose the baseline covariate $X$ has three categories. In a hypothetical study, 100 complete cases include 40 with $X=0$, 60 with $X=1$, and none with $X=2$. The dropout pattern includes 15 participants with $X=2$, and $Y_2$ is binary. The complete-case conditional event probabilities can be estimated for $X=0$ and $X=1$, but not for $X=2$, because the denominator for that stratum is zero.

The mixture for the dropout group would need the term

P(Y_2=1\mid X=2,C)\,P(X=2\mid D).

Here $P(X=2\mid D)=15/N_D$, where $N_D$ is the dropout group’s size, but the first factor is unobserved among complete cases. The restriction therefore does not provide an empirically estimated borrowed probability for this stratum. For a binary outcome, without an additional assumption or model, that conditional probability could range from 0 to 1. Reporting a precise value obtained by extending the $X=0$ or $X=1$ estimate to $X=2$ would conceal an extrapolation assumption.

The appropriate response is to report the lack of overlap, consider whether categories can be combined on substantive grounds, or use an explicit model and assess its extrapolation assumptions. Combining categories after seeing the results solely to make the calculation possible does not by itself justify the restriction.

Diagnostics, alternatives, and limitations

Several checks help reveal when the calculation is unstable or poorly supported:

Compare covariate-history support. Tabulate $X$ and relevant observed history $H$ by pattern. Look for dropout strata with no complete cases and for strata with very small counts.
Inspect conditional estimates. Very small complete-case denominators can make estimated conditional distributions highly variable, even when overlap is technically present.
Check the weighting calculation. Confirm that the weights are nonnegative, sum to one, and come from the dropout pattern being described. A weighted event probability must lie between 0 and 1.
Assess model extrapolation. With continuous covariates, compare the range and combinations of covariates among dropouts with those among complete cases. A model may be fitting outside the region supported by complete cases.
Keep the assumption visible. Agreement with observed-data summaries cannot establish equality of distributions for outcomes that are missing in $D$.

An unconditional complete-case restriction is a simpler alternative: it borrows one pooled distribution without conditioning on $X$ and $H$. It may be reasonable under a stronger assumption of equal distributions across patterns, but it can disregard important compositional differences. The conditional restriction permits covariate-related differences by borrowing within strata, at the cost of needing adequate support and more stable estimation in those strata. Other pattern-specific restrictions can make different assumptions about which observed patterns provide the borrowed distribution. None is selected by the observed data alone; the substantive assumptions and their implications should be made explicit.

A common mistake is to estimate conditional outcome probabilities among complete cases and then weight them by the complete cases’ covariate proportions. That calculation answers what distribution the complete cases have. To construct the distribution for $D$, use $P(X,H\mid D)$. A second mistake is to call the conditional restriction “automatic adjustment for covariates.” Adjustment changes the assumption but does not prove it: the missing outcomes remain unobserved, and conditional equality remains untestable from those outcomes.

Key takeaway: A covariate-conditional complete-case restriction borrows a conditional outcome distribution within baseline-covariate and observed-history strata. The distribution for the dropout pattern is then formed using that pattern’s own stratum weights. Overlap and the assumed conditional equality are essential; neither is guaranteed by the formula.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

State the conditional complete-case restriction using $X$, $H$, $C$, and $D$.
Complete cases have conditional event probabilities 0.10 and 0.50 in two $X$ strata. The dropout group is 70% in the first stratum and 30% in the second. What event probability does the restriction imply for that dropout group?
Why is a complete-case stratum with no members a problem when that stratum occurs among dropouts?
What is wrong with weighting stratum-specific probabilities by the complete cases’ covariate proportions when the target is the dropout distribution?

Solutions. (1) The restriction is $F_{Y_2\mid X,H,D}(y\mid x,h)=F_{Y_2\mid X,H,C}(y\mid x,h)$ in supported strata. (2) The implied event probability is $0.10(0.70)+0.50(0.30)=0.07+0.15=0.22$, or 22%. (3) There is no observed complete-case conditional distribution to borrow in that stratum; estimation requires an additional assumption or model-based extrapolation. (4) Those weights reconstruct the complete-case composition, not the dropout composition. The dropout distribution requires weights $P(X,H\mid D)$.

Complete-Case Restrictions With Baseline Covariates

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